Quotient space of a Hausdorff space is also a Hausdorff space

I would go about it this way (letting $Y$ represent the quotient space $X/\sim$):

• First, use the fact that $\pi\circ s$ is the identity on $Y$ (so is one-to-one) to prove that $s$ is one-to-one.
• Next, letting $Z$ be the image of $s,$ note that $Z$ is a subspace of $X,$ so is Hausdorff.
• Finally, show that $\pi\restriction Z$ is the inverse of $s.$ Since both $s$ and $\pi\restriction Z$ are continuous, that means that $s$ is a homeomorphism from $Y$ to $Z,$ so since $Z$ is Hausdorff, so is $Y.$